Frank Jenkowww.ipp.mpg.de/~fsj/gene

Max-Planck-Institut für Plasmaphysik, GarchingUniversität Ulm

Kinetic-scale turbulence in laboratory and space plasmasCambridge University, 20 July 2010

GraspingPlasma Turbulence

Fundamentals:Where do we stand?

The plasma turbulence challenge

The ultimate answer to Life,the Universe and Everything is...

42

How about our neighbors?The fluid turbulence challenge

Turbulence as a local cascade in wave number space…

The Navier-Stokes equation in action

kE

ηk k

energy flux

fk

driverange

dissipationrangeinertial

range

Spectral energy balance

E = v2 d3x = E(k) dk⌠⌡

∞12V

⌠⌡0

Kolmogorov’s theory from 1941

K41 is based merely on intuition and dimensional analysis –it is not derived rigorously from the Navier-Stokes equation

Key assumptions:

• Scale invariance – like, e.g., in critical phenomena• Central quantity: energy flux ε

E(k) = C ε2/3 k-5/3

This is the most famous turbulence result: the “-5/3” law.However, K41 is fundamentally flawed: Scale invariance is broken!

Key open issues: Inertial range• Is the inertial range physics universal (for Re → ∞)?

• If so, can one derive a rigorous IR theory from the NSE?

• How should one, in general, handle the interplay betweenrandomness and coherence?

Example: Trapping of tracers in vortex filaments

Note:

The observed deviations from self-similaritycan be reproduced qualitatively by relativelysimple vortex models.

See, e.g., Wilczek, Jenko & Friedrichs, PRE 2008Bife

rale

et a

l. 20

05

Key open issues: Drive range• Often, one is interested mainly in the large scales. Here,

one encounters an interesting interplay between linear(drive) and nonlinear (damping) physics. – Is it possibleto remove the small scales?

• Candidates: LES, Dynamical systems approach etc.

Ore

llano

and

Wen

gle

2001

How about us?Fundamental open issues

in plasma turbulence research

Appropriate field equations

X = gyrocenter positionV� = parallel velocityµ = magnetic moment

Advection/Conservation equation

The gyrokinetic equations in action

The kinetic version of reduced MHD…

Microinstabilities driving plasma turbulence

Doyle et al.

Some fundamental open issues

• How do the various microinstabilities saturate?

• How is the (free) energy distributed and dissipated?

• How useful is the concept of an inertial range?

• What is the role of sub-ion-gyroradius scales?

What do we really knowabout nonlinear saturation?

The drift wave / zonal flow paradigm

Many years of studying ITG turbulence(mostly using adiabatic electrons)

led us to think thatthe physics of nonlinear saturation is

synonymous with zonal flow shearing.

Is this view really correct?

Historical parallels

Gaul is entirely occupiedby the Romans.

Well, not entirely...

Trapped electron mode turbulence

Pure TEM turbulence simulations [Dannert & Jenko, PoP 2005]:

- In the drive range, nonlinear and linear frequencies are identical

- In the drive range, there is no significant shift of cross phases w.r.t. linear ones

– No dependence of transport level on presence or absence of zonal flows

Zonal flows suppressed

linear

nonlinear

ZF / Non-ZF regimes

Ernst et al., PoP 2009

ExB shearing rates exceed thegrowth rate only for ηe < 1

For mainly temperature gradientdriven TEM turbulence, ZFs(and GAMs) are relatively weak

Thus, in a wide region of parameterspace, the standard drift-wave / ZFparadigm does not hold

� Both weak and strong turbulence theories suggest that the ExB nonlinearity can be represented by a coherent part and a random noise part

� and are fluctuating quantities; minimizing the model error , we obtain

Theory-motivated statistical analysis

Low-ky drive range: large transport contributions, but small random noise; here, one finds:

Saturation of TEMs: “eddy damping”

~ky2

Merz & Jenko, PRL 2008

This is in line with various theories, including Resonance Broadening Theory (Dupree), MSR formalism (Krommes), Dressed Test Mode Approach (Itoh).

Implications for transport modeling

NL frequency spectra

Results motivate use of thefollowing quasilinear model: GENE sim’s

QL model

In situations where ITG modesand TEMs compete, they cancoexist, and there can be non-trivial nonlinear interactions

Merz & Jenko, NF 2010

Dissipation & cascadesin plasma turbulence

D.R. Hatch, P.W. Terry, W.M. Nevins, F. Merz & FJ

Turbulence in fluids and plasmas –Three basic scenarios

Saturation via damped eigenmodes

Excitation of damped eigenmodes

Using GENE as a linear eigenvalue solver to analyzenonlinear ITG runs via projection methods, one finds…

unstableeigenmode

least damped eigenmode

ky=0.3drive range

strongly dampedeigenmodes

(fine-scale structurein v║ and z)

Energetics in wavenumber space

Damped eigenmodes are responsible forsignificant dissipation in the drive range.

Some energy escapes to high k

Resulting spectrum decays exponentially @lo k, asymptotes to power law @hi k

Multiscale wavenumber spectra

Poloidal wavenumber spectraof density fluctuations for pureTEM / ITG / ETG turbulence

Poloidal wavenumber spectraof density fluctuations for mixed

TEM / ITG – ETG turbulence

NL (!) ETG drive

L ETG drive

Universality? T. Görler & FJ, PoP 15, 102508 (2008)

Poloidal wavenumber spectraof density fluctuations for pureTEM / ITG / ETG turbulence

Poloidal wavenumber spectraof density fluctuations for mixed

TEM / ITG – ETG turbulence

T. Görler & FJ, PoP 15, 102508 (2008)

ky spectra at kx=0 are steeper

kx spectra differ

What is the role ofsub-ion-gyroradius scales?

High-k turbulence simulationsETG turbulence can induce significant electron heat transport:

is possible (Jenko et al., PoP 2000;Dorland et al., PRL 2000)

For comparison: (Cyclone base case)

Confirmed, e.g., by (Idomura et al., NF 2005) and (Nevins et al., PoP 2006).

Latter paper: A prefactor of ≤10 is sufficient to explain certain experiments.

Coexistence of ITG and ETG modes

ITG/TEM/ETG turbulence: Large fraction ofelectron heat transport is carried by electron scales.

[Görler & Jenko, PRL 2008]

Ion-scale transport muchlarger than in experiments.

box size: ~64 ρi resolution: ~2ρe

Physics of H-mode barriers

• Strong ExB shear flows thought to suppress long-wavelength turbulence

• Ion heat transport close to neoclassical, but other transport channels

remain anomalous

• What sets the residual electron heat transport?

Some candidates for setting the residual electron heat transport

• Paleoclassical transport (theoretical foundations are disputed)

• Residual long-wavelength turbulence (not ITG)

• High-wavenumber turbulence (e.g., ETG)

This possibility will be investigated by means of gyrokinetic simulations…

Special case: The H-mode edge

ASDEX Upgrade #20431 (ρpol = 0.98)

separatrix

ρpol = 0.98

Edge transport barrier region:

• kyρs < 0.1 → ITG mode

• kyρs ~ 0.15 → microtearing mode

• kyρs > 0.2 → ETG mode

Told et al., PoP 2008

Linear stability of edge ETG modes

In the core, linear growth rates tend to peak at kx = 0;here, they peak at large kx values.

Transport spectrum from GENE

In contrast to the linear growth rate spectrum,the transport spectrum peaks at low kx values.

Electron heat transport in edge barriers

ETG turbulence is able to explain the residualelectron heat transport in H-mode edge plasmas.

AUGJenko et al., P

oP16, 055901 (2009)

More on this topic:Next week

Summary and outlook

• Recent surprises and advances concerning thenonlinear saturation, dissipation, and multi-scaleproperties of plasma turbulence(see also the posters on nonlocal effects by S. Brunner & T. Görler)

• More investigations targeted at improving ourunderstanding of fundamental issues in plasmaturbulence are certainly called for

• In the end, all of this is bound to have importantpractical consequences concerning the efficienttransport modeling of ITER plasmas